LOCAL FIELDS AND TREES

7

and attains the same values; that is the set { d(x, y) : x, y

E

X} is the same for all

distances defined as above, independently of the choice of the origin o.

These observations prove that every element of

Aut(X)

acts continuously on

n.

This is clear for every rotation because a rotation is an isometry with respect to

an equivalent metric. Since every automorphism can be written as the product of

rotations and a single translation of step one, it is sufficient to show that, with

respect to the metric defined by fixing a vertex, say o, a step one translation along

a geodesic containing o is continuous. This is easily seen with a simple geometric

argument (a ball is either contracted in a smaller ball or dilated into a finite union

of balls).

THE TREE OF A LOCAL FIELD

We refer to [7] for definitions and basic properties of a local field. We shall also

follow if not otherwise stated the symbols and terminology of [7]. Thus we shall

denote a local field by F. We recall here that on F is defined the modular function

!alp

which satisfies the multiplicative property

and the ultrametric inequality

The modular function is nonnegative and attains the value zero only at 0.

It

defines therefore a metric on F. The ultrametric property implies that closed balls

of a given radius form a partition of F. We shall presently describe these balls.

Recall that the unit ball

0

={a

E

F:

!alp~

1}

is compact and open.

The ultrametric inequality implies that 0 is a ring. As such it contains a unique

maximal ideal, namely

P

={a

E

F:

!alp

1}.

We shall also use the hotation

ox ={a

E F:

!alp=

1},

so that

0

=

P

U

0

x .

Let p be a generator of P (we depart in this case from the notation of [7], where

the symbol

1r

is used in place of p). Then !PIP

= l,

where

q

is the order of the

q

finite field 0 /P, and P

=

pO.

The compact open sets

form a basis of neighborhoods of 0 which

fork-+

-oo exaust

F.

7

and attains the same values; that is the set { d(x, y) : x, y

E

X} is the same for all

distances defined as above, independently of the choice of the origin o.

These observations prove that every element of

Aut(X)

acts continuously on

n.

This is clear for every rotation because a rotation is an isometry with respect to

an equivalent metric. Since every automorphism can be written as the product of

rotations and a single translation of step one, it is sufficient to show that, with

respect to the metric defined by fixing a vertex, say o, a step one translation along

a geodesic containing o is continuous. This is easily seen with a simple geometric

argument (a ball is either contracted in a smaller ball or dilated into a finite union

of balls).

THE TREE OF A LOCAL FIELD

We refer to [7] for definitions and basic properties of a local field. We shall also

follow if not otherwise stated the symbols and terminology of [7]. Thus we shall

denote a local field by F. We recall here that on F is defined the modular function

!alp

which satisfies the multiplicative property

and the ultrametric inequality

The modular function is nonnegative and attains the value zero only at 0.

It

defines therefore a metric on F. The ultrametric property implies that closed balls

of a given radius form a partition of F. We shall presently describe these balls.

Recall that the unit ball

0

={a

E

F:

!alp~

1}

is compact and open.

The ultrametric inequality implies that 0 is a ring. As such it contains a unique

maximal ideal, namely

P

={a

E

F:

!alp

1}.

We shall also use the hotation

ox ={a

E F:

!alp=

1},

so that

0

=

P

U

0

x .

Let p be a generator of P (we depart in this case from the notation of [7], where

the symbol

1r

is used in place of p). Then !PIP

= l,

where

q

is the order of the

q

finite field 0 /P, and P

=

pO.

The compact open sets

form a basis of neighborhoods of 0 which

fork-+

-oo exaust

F.